Under the paradigm of Edge Computing (EC), a Network Operator (NO) deploys computational resources at the network edge and let third-party Service Providers (SPs) run on top of them, as tenants. Besides the clear advantages for SPs and final users thanks to the vicinity of computation nodes, a NO aims to allocate edge resources in order to increase its own utility, including bandwidth saving, operational cost reduction, QoE for its users, etc. However, while the number of third-party services competing for edge resources is expected to dramatically grow, the resources deployed cannot increase accordingly, due to physical limitations. Therefore, smart strategies are needed to fully exploit the potential of EC, despite its constrains. To this aim, we propose to leverage service adaptability, a dimension that has mainly been neglected so far: each service can adapt to the amount of resources that the NO has allocated to it, balancing the fraction of service computation performed at the edge and relying on remote servers, e.g., in the Cloud, for the rest. We propose EdgeMORE, a resource allocation strategy in which SPs express their capabilities to adapt to different resource constraints, by declaring the different configurations under which they are able to run, specifying the resources needed and the utility provided to the NO. The NO then chooses the most convenient option per each SP, in order to maximize the total utility. We formalize EdgeMORE as a Integer Linear Program. We show via simulation that EdgeMORE greatly improves EC utility with respect to the standard where no multiple options for running services are allowed.